On the effectiveness of a rod-like distributed piezoelectric controller in preventing the Hopf bifurcation of the visco-elastic Beck’s beam

The linear stability of a piezo-electro-mechanical (PEM) system subject to a follower force is here discussed. The mechanical subsystem is constituted by a linear visco-elastic cantilever beam, loaded by a follower force at the free end. It suffers from the Hopf bifurcation, whose critical load is strongly affected by damping, according to the well-known Ziegler’s paradox. On the other hand, the electrical subsystem consists of a distributed array of piezoelectric patches attached to the beam and connected to a properly designed second-order analog circuit, aiming at possibly enhancing the stability of the PEM system. The partial differential equations of motion of the PEM system are discretized by the Galerkin method. Linear stability analysis is then carried out by numerically solving the associated eigenvalue problem, for different significant values of the electrical parameters. A suitable perturbation method is also adopted to detect the role of the electrical parameters and discuss the effectiveness of the controller.